1. Convective throughout deliver heat from core to surface purely by convection.

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1 6/30 Post Main Sequence Evolution: Low-Mass Stars 1. Convective throughout deliver heat from core to surface purely by convection. 2. Convection mixes the material of the star is the material carries the heat. 3. Once the H is gone in the center of the star, it is gone everywhere in the star. 4. The star is now all He and can t get hot enough to fuse He into other elements. 5. Once the H is gone, fusion stops. 6. No longer outward radiation pressure; the star begins to gravitationally contract again. 7. Eventually, this contraction stops due to Pauli Repulsion. (Aside: Pauli Exclusion Principle To explain the structure of the periodic table of the elements why different elements have different chemistry hypothesized that electrons could not share the same quantum state. Also true of protons, neutrons, and neutrinos among other particles.) Further contraction of the star would cause electrons to occupy the same quantum states can t happen. Matter in this state is called degenerate electrons occupy all available quantum states. 8. The star is now about the size of the Earth and has become quite hot because of its contraction. 9. It is a white dwarf. 10. The white dwarf sits in space radiating its heat away and eventually becomes a black dwarf. More Massive Stars: Either the star has a radiative core and convective envelope (medium-mass stars such as the Sun) or the star has a convective core and radiative envelope (most massive stars). In either case, there is no mixing between the material in the core and the material in the envelope. Eventually, the hydrogen is used up in the core but, when this happens, there is still H in the envelope.

2 When the H is used up in the core, the core begins to contract. When the core contracts, it gets hotter this causes H fusion into He to start in the envelope. The outflow of energy from this region, because of its low density, causes the envelope to expand. The star expands and cools becomes a red giant. In the lower mass stars among the medium mass star, the core will never get hot enough to fuse He into heavier elements. The outflow of energy in the envelope eventually expels the envelope into space, leaving behind the core of the star. The core of the star contracts until stopped by Pauli repulsion becomes a white dwarf. Our own Sun will get hot enough to fuse He into heavier elements carbon and oxygen. It will spend a short amount of the time on the He main sequence. The Sun will not get hot enough to fuse carbon and oxygen into heavier nuclei. The Sun will also expel its envelope and end up as a white dwarf made of carbon and oxygen. More massive stars will be able to fuse carbon and oxygen into heavier elements. More and more massive stars will eventually form white dwarfs of heavier and heavier elements. Until we get to iron that s the next chapter. We can test these ideas by examining star clusters. Two types of star cluster: 1. Open few stars a few 10's to a few 1000's stars tend to be young where stars are born size about 10 pc irregular shape 2. Globular Star Cluster Many stars 100's of thousands to millions of stars. stars tend to be old may be the oldest structures in the universe sizes 20 to 30 pc name from their spherical shape like a globe

3 Assume that all the stars in a cluster were created at about the same time. We should see different mass stars in clusters at different stages of evolution. Expect to see for an open cluster: 1. No low-mass stars on the main sequence not enough time to evolve that far. 2. Only the most massive stars very few will have evolved off the main sequence. 3. Draw HR diagram for an open cluster. 4. The turn-on point, where stars are just evolving on to the main sequence can be used to deduce the age of the cluster. Expect to see for a globular cluster. 1. Even the lowest mass stars will have had time to evolve onto the main sequence. 2. The massive stars will have had time to evolve off the main sequence. 3. Draw HR diagram for a globular cluster. 4. Turn-off point tells us about the age of the cluster. Chapter 10 The Deaths of Stars We have already discussed the deaths of less massive stars end as white/black dwarfs. Here we discuss the end states of the most massive stars. Chandrasekhar discovered that a white dwarf can be no more massive than 1.4 solar masses if it is, not even Pauli repulsion can prevent further contraction. We will now discuss what happens to stars whose end states have masses greater than the Chandrasekhar limit. The most massive stars, after leaving the main sequence, will continue to fuse elements until they get to iron. Iron is the most stable of nuclei produced inside a star no energy can be generated from fusing iron into heavier elements. When the core of the star becomes iron, fusion stops.

4 In a fraction of a second, the core collapses. The core rebounds like a compressed spring that produces an explosion that tears the star apart supernova. A supernova can be brighter than the galaxy of which it is apart. A type II SN such as this one can be as bright as 600 million Suns. One nearby can be bright enough to shine out in the daytime. A SN within 50 ly of the Earth could destroy all life here on Earth. What is left at the center? Depends on the mass of the star. If the main sequence mass is less than 40 solar masses, the remnant will be between 1.4 solar masses and 2 to 3 solar masses. Since the mass is greater than the Chandrasekhar limit, not even the Pauli repulsion of the electrons can prevent further contraction. The electrons are forced into protons, converting them to neutrons. We end up with a ball of neutrons with a few electrons and protons floating around that continues to contract. But neutrons also must obey the Pauli principle object contracts until further contraction would cause neutrons to occupy the same quantum states. We end up with a neutron star. Characteristics of a Neutron Star Mass about 2 solar masses radius about 10 km like a huge nucleus density is the same as the density of an atomic nucleus because they start big with a slow rotation, the end up small with a rapid rotation about 30 times a second. material of the neutron star behaves like a superconductor magnetic field of the neutron star is trapped in the superconducting matter and rotates around with the star We have found neutron stars first were pulsars that

5 sent rapid pulses of the light in our direction. The magnetic field axis in a neutron star will probably be off the spin axis case here on Earth. Charged particles trapped in the magnetic field send beams of light along the axis of the magnetic field. As the neutron star spins, the beam of light sweeps around. When pointed at Earth, it sends a pulse our way. We see a series of pulses, say, 30 times a second. This is the lighthouse model of a neutron star/pulsar. We can also detect neutron stars in close binary systems neutron star is close enough to its companion to bleed mass off of it. As the matter loses gravitational energy, it converts it to x rays we can detect the x rays. We know it s a neutron star because we know what sort of x ray spectrum we should see from a neutron star. If it is a not close binary system, and we calculate the mass of the unseen companion and if it has the mass of a neutron star, we know it s a neutron star. Gravitational Lens Effect If a neutron star comes between us and distant object, it will cause the distant object to increase in brightness. How much and how long tells us the nature of the impeding object and, with the right properties, it will be a neutron star. Astrometric Binary Star Only see one of the two stars. Analyzing the system will allow us to find the mass of the unseen companion. If that mass is in the range of neutron star masses, it is a neutron star. What if the final mass of the remnant is greater than 2 to 3 solar masses (main sequence mass of more than 40 solar masses)? Not even the Pauli repulsion of the neutrons can prevent further contraction. Gravity takes over and the object contracts without stopping. What goes on until the object gets to be atomic size or so is governed by the general theory of relativity GR. Special Theory of Relativity SR Michaelson-Morely Experiment Hypothesis: electromagnetic waves travel through a medium called the ether.

6 M-M set about finding the motion of the Earth relative to the ether. Compared the speed of a light signal along perpendicular paths. In the figure, suppose that the ether is traveling horizontally toward the right. Light from the source S travels toward mirror M 1, which is a half-silvered mirror. Some of the light passes through toward mirror M 3, where it is reflected back. Some of the light is reflected toward mirror M 2, where it is reflected back. The two beams of light are recombined at mirror M 1, and observed by the observer. If the speed of light is different along the two paths, the observer will see the effect of interference between the two beams. If the ether flows, say, horizontally the right, the horizontal light beam will take less time to complete the trip than the vertical one will. One should see interference and from the amount of interference, the speed of the ether can be deduced. No interference was seen. Found no difference in the speed of light in the two different directions. Two Postulates for the Special Theory of Relativity: 1. The laws of physics are the same in all unaccelerated frames of reference. 2. All observers, regardless of their frame of reference, will measure the same value for the speed of light. Consequences. 1. Simultaneity Two events that are simultaneous in one frame of reference are not necessarily simultaneous in any other frame of reference. Note that, if two events occur simultaneously at the same point in space, they will be simultaneous in all frames. If two events are simultaneous in one frame of reference, they are not necessarily simultaneous in any other frame of reference. Consider the light train in the picture below:

7 Fast train travels through a station without stopping at the platform. We have two observers: T on the train in the middle of the train and P on the platform directly across from T (at the time shown). At the instant shown, two lighting bolts hit the front and rear of the train simultaneously in the frame of P. We have set up the problem so that the bolts strike simultaneously in the platform frame; we will show that they do not strike simultaneously in the train frame. What P sees with regard to the lightning bolts: 1. P sees the light from the bolts arrive at his position at the same time. 2. P sees that the light from the two bolts has traveled the same distance half the length of the train. 3. P measures the speed of the light from both bolts to be the same. 4. P sees the light signals from the two bolts have traveled the same distance in the same time at the same speed, and concludes that the two bolts struck the train at the same time. They are simultaneous in his frame. What P observes about T: 1. P sees T moving toward the light signal from the front bolt and away from the light signal from the rear bolt. 2. P sees the light from the front bolt arrive at T s location before the light from the rear bolt. 3. Since this sequence of events occurs at the same point in T s frame, this sequence of events is the same in both frames.

8 What T sees about the lightning bolts: 1. T sees the light from the front bolt arrive before the light from the rear bolt. 2. T sees that both light signals have traveled the same distance half the length of the train. 3. T measures the speed of the light signals to have the same value (2 nd postulate). 4. T sees the light signals from the two bolts travel the same distance at the same speed. But, since the light from the front bolt arrives first, he concludes that the front bolt occurred first. The bolts aren t simultaneous in T s frame. In Newtonian physics, time is absolute and the same in all frames. But Einstein made the speed of light absolute. Something had to give, and it was absolute time (and, as we will see, space). This suggests that time passes at different rates in the two frames. Time Dilation Consider the light clock. A base containing electronics, a light bulb, and a detector. A pulse is emitted from the light bulb, travels to a mirror, and is reflected back to the detector. The electronics then generates the next pulse. Place an ordinary clock at the location of the light light clock and synchronized with it. Since they are located at the same point in space, they will be synchronized in all frames. We will accelerate ourselves up to a large speed moving to the left. In this frame, the clock will move to the right. Draw the light clock as observed in this new frame.

9 In the new frame, the clock travels from left to right with speed V. In this frame, the light pulse travels a greater distance than it does in the rest frame of the clock. But it does so at the same speed because postulate 2 tells us that speed of light is the same in both frames. It takes longer for the light pulse to get from light bulb to detector in the moving frame. The clock ticks more slowly in the moving frame. Time passes more slowly in the moving frame. Time Dilation: In this formula, the t s are time intervals, one in the rest frame of the clock (the frame in which the clock is at rest) and one in the frame in which the clock is moving. This result has been tested experimentally by accelerating unstable particles called muons to nearly the speed of light. The fast moving muons have longer lifetimes than muons at rest. There is numerical agreement within uncertainty with Einstein s prediction. In fact, this is the reason that muons form about 75% of the secondary

10 cosmic radiation at the surface of the Earth. Their lifetimes are not long enough for them to reach the surface of Earth without time dilation. Showers of muons are created in the upper atmosphere, roughly 100 km up. They travel at nearly the speed of light. They have a lifetime of 2.2 s or 2.2 millionths of a second. Without time dilation, the muons would travel 300,000,000 m/s s = 660 m. Due to time dilation, the moving muons last much longer and can reach the surface of the Earth before they decay. Length Contraction moving rods are measured to be shorter than when measured at rest. Think about the muons. Consider those created in the upper atmosphere at nearly the speed of light. In their own rest frame they don t last very long. In their rest frame, the surface of the Earth is rushing up to meet them at nearly the speed of light. Since they only last for 2.2 microseconds in their rest frame, how does the surface of the Earth reach them before they decay? In their frame, the distance between them and the surface of the Earth is shortened Length Contraction: So far we have been discussing kinematics how coordinates and times change. 2. Time Dilation Moving clocks run slow. where t is the time between ticks in the moving clock, t 0 is the time between ticks for the clock at rest, v is the speed of the clock, and c = speed of light. Has been tested: 1. Atomic clocks, synchronized, and then one sent on a plane trip for 24 hours, brought back disagree just as shown above. 2. GPS takes this into account. 3. Muons are unstable particles with a lifetime of 2.2 s. They last longer when their lifetimes are measured while they are in motion. The time dilation relation predicts, within experimental error, what is observed for the lifetime of the moving muons.

11 Length Contraction Muons are created in the upper atmosphere due to collisions between cosmic rays and molecules of air. These muons get to the surface of the Earth and form about 75% of the background radiation at the surface. But traveling a the speed of light for 2.2 s, they would only be able to travel a few hundred meters and wouldn t make it to the surface of the Earth. They get here due to time dilation the rapidly moving muons last long enough to make to the surface. Consider the rest frame of the muons in this frame, they last for only their 2.2- s lifetimes with the Earth rushing up to meet them at near the speed of light. The Earth is only going to move a few hundred meters in the 2.2 s how does it reach the muons before they decay? The distance from the Earth to the muons, in the muon rest frame, has shrunk or contracted to a few hundred meters. Moving rods are shorter. where L 0 is the rest length of the rod and L, the moving length. General Theory of Relativity GR Two Postulates of the General Theory of Relativity: 1. The laws of physics are the same in all frames of reference. 2. The principle of equivalence. Three statements of the principle of equivalence: 1. The inertial mass (the one that appears in Newton s second law, F = ma) is the same as the gravitational mass (the one that appears in Newton s law of gravity). 2. No experiment can be done in a closed elevator that will distinguish between being in free fall in a gravitational field or moving with constant velocity in gravity-free space. 3. No experiment can be done in a closed elevator that will distinguish between being at rest in a gravitational field or having a uniform acceleration in gravity-free space.